While many base stations do operate on AC wall current (but some do not), the rest of what you have described is pretty much incorrect. And mobiles, as a general rule, do not hear any better than base stations, it is totally radio specific. The sensitivity, phase noise, dynamic range, noise figure, image rejection, etc are just a few of the variables that end up defining radio performance.
For a base station that operates from an AC wall outlet and is transistor based; the vast majority of the time the AC wall current is brought into the radio and does go through a transformer, unless the radio uses a switching supply. However, after the transformer the stepped down AC (still AC) is rectified to DC. It is then (typically) filtered and regulated to about 13.8 VDC before going to the electronics of the radio. This is essentially identical to what is done with a mobile and an external power supply, except in the case of a mobile there are two pieces of hardware, and in a base station the two are often combined in one case. Let me say it again, in general the exact same processes are in place for a mobile working on an external supply and a base station, the base station just combines it all in one box.
The reason base stations often regulate to about 13.8 VDC is because that is the nominal design voltage for many CB’s. Several, probably many, base stations use the same electronics and PC boards as some mobiles, sometimes the only difference is that the base station version has a built in regulated power supply (generally on a separate PC board from the main electronics, sometimes in a physically separated section of the chassis), at other times the PC board is slightly different to accommodate the added features sometimes found on the base versions.
While a mobile might be able of operation on 14.0 to 14.5 VDC at the radio input (many will go to 15 VDC) there is generally no good reason to do so. Most mobiles are designed for an optimal power supply of 13.8 VDC. By cranking up the voltage you might get another Watt or fraction of a Watt of transmit power, a small and generally useless increase (there often is no other operator detectable change), at the cost of the radio generally running hotter. So, why crank up the supply voltage?
T!